Included-angle ray, closing side, and dropped altitudeLive current inputs
Math & Statistics
Triangle Step-by-Step Calculator
Construct and solve a triangle from two sides and their included angle, showing the Law of Cosines substitution, remaining angles, area, projection, and dropped altitude.
Solved opposite side a-
Solved angle B-
Solved angle C-
Area from SAS-
Altitude to side c-
Horizontal projection-
Perimeter-
Angle-sum check-
CONSTRUCTION SEQUENCE
Included-angle ray, closing side, and dropped altitude
The diagram starts with base c and angle A, places side b on the ray, then closes side a. Projection and altitude are dimensioned separately.
STEP-BY-STEP DERIVATION
From the included angle to every remaining element
Each row records the formula, current substitution, result, and why that step is valid.
| Step | Purpose | Formula | Current substitution | Result | Cross-check |
|---|
SAS INPUT
Confirm that the angle lies between the two entered sides
- Identify the included angle shared by sides b and c.
- Use one unit for both side lengths.
- Enter an interior angle strictly between 0° and 180°.
- Solve side a before using a sine ratio.
- Compare the drawn projection with the sign of cos A.
WHY ORDER MATTERS
SAS avoids the ambiguous sine case when solved in the right sequence
The Law of Cosines uses exactly the known sides and their included angle, so the closing side is unique.
After all three sides are known, the remaining angles can be recovered and reconciled to 180°.
SAS CONSTRUCTION LOGIC
Close the triangle before using the Law of Sines
Two sides with their included angle define one triangle. The Law of Cosines solves the closing side first; the Law of Sines then recovers a remaining angle without the SSA ambiguity.
Detailed calculation process and general formulas
a² = b² + c² - 2bc cos Aa = √(b² + c² - 2bc cos A)sin B / b = sin A / aC = 180° - A - BK = ½bc sin ASymbols, meanings, and units
- b,c
- entered sides enclosing angle Alength
- A
- entered included angledegrees
- a
- closing side opposite Alength
- h
- altitude from the endpoint of side b to clength
- K
- triangle arealength²
CONSTRUCTION INSIGHT
Projection explains acute and obtuse included angles
The altitude splits side b into orthogonal components.
Along-base projection
-b cos A is positive for acute A and negative when the ray points behind the base.
Perpendicular height
-b sin A sets the altitude and therefore the area.
Closing side
-The distance between the two endpoints is solved by the cosine law.
Decision takeaway: When A approaches 0° or 180°, the geometry becomes nearly flat and increasingly sensitive.
FIELD CONSTRUCTION NOTES
Measurements to preserve with an SAS record
- Which vertex carries angle A
- Instrument angular precision
- Whether lengths are slope or horizontal distances
- Unit and datum
- Rounding applied after solution
Applied decisions
Two included angles with the same sides
Acute included angle
Side b projects forward along base c.
What the result clarifies: The altitude is positive and the closing side can be shorter than either entered side.
Obtuse included angle
The projection of b falls behind the A vertex.
What the result clarifies: The closing side grows and the area still follows one-half bc sin A.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This page assumes Euclidean SAS data with the entered angle lying between sides b and c. Survey reductions, spherical triangles, measurement error, and ambiguous non-included-angle cases require different treatment.
Triangle Step-by-Step Calculator | SAS Construction and Altitude FAQ
Why solve side a first?
SAS directly supports the Law of Cosines and avoids using an unknown opposite pair.
Can the projection be negative?
Yes. For an obtuse included angle, b cos A points opposite the positive base direction.
Why does area shrink near 180°?
The sine of the included angle approaches zero, so the altitude collapses.
Is there a second SAS triangle?
No. Two sides and their included angle determine one shape up to reflection.